2021 JMC 10 Problems/Problem 3

Problem

A group of $8$ people are either honest or liars, where honest people always tell the truth and liars always lie. People $P_1,P_2, \cdots, P_8$ stand in a line, and person $P_i$ calls $P_{i+1}$ a liar where $P_1 = P_9.$ Out of these eight people, how many liars are there?

$\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 4 \qquad\textbf{(E) } 7$

Solution

If one $P_i$ calls $P_{i+1}$ a liar, then only one of $\{P_i,P_{i+1}\}$ can be a liar. If both are liars, we have a contradiction, because a liar would call another liar a truth teller. Likewise, $\{P_i,P_{i+1}\}$ cannot both be truth tellers. We must have an alternating sequence of truth tellers and liars, so there are $\tfrac{8}{2} = 4$ liars.

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2021 JMC 10 Problems/Problem 3

Problem

Solution